Quarters, Dimes and Linear Combinations
Lesson 7 of 14
Objective: SWBAT model a combination of quarters and dimes as a system of linear equations and solve by elimination.
I plan for this Warm up to take about 5 minutes for the students to complete and about 5 minutes to discuss the solutions found. Most of the students draw coins on their paper or make tables to work this problem. This is a concrete example that I allow students to solve using any method. Money is a concept that students understand, and a good concrete example for students. After students complete the warm up, I question them about the more abstract idea of writing algebraic equations to represent the situation. I use this lesson as an introduction to solve a system of equations using linear combinations, which is I also refer to as elimination.
I question students after the warm up:
- How did you find the number of quarters and dimes needed to represent $1.45?
- What 2 math operations do you use to find the money total and number of quarters and dimes?
- How many possible solutions are there to this problem?
- What are the 2 constraints of this problem?
- What 2 equations could you write to represent this problem as a system of equations based off of those constraints?
- How can you solve this system of linear combinations by Elimination?
I model showing students how to write the algebraic equations in the video below.
In this think-pair-share activity, I provide the students time to work in pairs to solve the system of equation word problems using elimination. After going over the warm up, I want students to be able to recognize the difference between these Linear Combination problems and the word problems we previously solved involving increasing and decreasing situations. I have focused several previous lessons on distinguishing between these two types of problems. In the increasing and decreasing problems, students should recognize an initial condition, and a constant rate of change. In linear combinations, there are two linear situations that can be defined with variables.
There are 4 word problems for students to solve. I purposely plan for all of the problems to be similar, so students can learn these type of problems from repeated reasoning. I use mathematical practice 7 and 8, to look at the repeated reasoning and the same structure.
I do specify students to write the equations of these problems to introduce solving a system of equations using elimination. After I present all of the different methods to solve a system of equations with practice, I give the students a summative assessment at the end of the unit. I allow students to use any method of their choice on the summative assessment. I teach my students that if a method is not specified, they may use any method of their choice as long as they support it with the math and reasoning.
I want students to be able to recognize the difference between a word problems that describe an increasing or decreasing situation, and, a word problems that describe a linear combination. So, I use this Exit ticket to summarize the differences, and to answer what is found by linear combinations. Why are linear combinations useful?